Digital Signal Processing Reference
In-Depth Information
6.2.1.2 Noise Level Estimation
The final step before applying equation (6.6) is to know the values of
σ
j
,
at each level and orientation or to estimate them from the observed noisy data.
For a stationary additive Gaussian noise
σ
and
, this can be done quite easily. For other
noise models, the picture becomes less clear, although appropriate strategies can be
developed for some noise models, as we will see later for the Poisson case.
ε
Variable
ε
Is Additive White Gaussian
To estimate
, the usual standard deviation of the data values is clearly not a good
estimator, unless the underlying signal
X
is reasonably flat. Donoho and Johnstone
(1994) considered estimating
σ
in the orthogonal wavelet domain and suggested a
robust estimator that is based only on the wavelet coefficients of the noisy data
Y
at the finest resolution level. The reason is that the wavelet coefficients of
Y
at the
finest scale tend to consist mostly of noise, whereas the wavelet coefficients of
X
at that scale are very sparse and can be viewed as outliers. Donoho and Johnstone
(1994) proposed the estimator
σ
median(
w
1
−
w
1
)
)
σ
=
˜
MAD(
w
1
)
/
0
.
6745
=
median(
/
0
.
6745
,
(6.9)
where MAD stands for the median absolute deviation and
w
1
are the orthogonal
wavelet coefficients of
Y
at the finest scale. For 2-D images, the preceding estimator
is to be applied with the diagonal subband of the 2-D separable orthogonal wavelet
transform.
We now turn to the estimation of
σ
j
,
. As the noise is additive, we have
η
j
,,
k
:
=
ε,ϕ
j
,,
k
, and it is easy to see that
σ
ϕ
j
,,
k
.
σ
j
,
=
˜
˜
(6.10)
If the atoms in the dictionary
all have an equal unit
2
norm, then obviously,
σ
j
,
=
2
norms were known
analytically, as is the case for the curvelet tight frame (see Section 5.4.2.2). But if
these norms are not known in closed form, they can be estimated in practice by
taking the transform of a Dirac and then computing the
˜
σ,
∀
˜
(
j
,
). This formula is also easily implemented if the
2
norm of each subband.
σ
j
,
consists of simulating a data
set containing only white Gaussian noise with a standard deviation equal to 1 and
taking the transform of this noise. An estimator of
An alternative obvious approach to estimate
σ
is obtained as the standard
j
,
deviation of the coefficients in the subband (
j
,
).
Variable
ε
Is Additive Colored Gaussian
Suppose now that the noise in the original domain is Gaussian but correlated. If
its spectral density function happens to be known, and if the transform atoms
ϕ
j
,,
k
correspond to a bank of bandpass filters (as is the case for wavelets and second-
generation curvelets), the standard deviation
σ
j
,
can be calculated explicitly from
the noise spectral density function and the Fourier transform of the atoms
ϕ
j
,,
k
(Papoulis 1984),
S
(
)
ˆ
)
2
d
j
σ
,
=
ν
ϕ
j
,,
k
(
ν
ν
,
